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Besicovich proved a long time ago that a straight line segment of fixed length could be rotated 360 degrees within a subset S of the Euclidean plane such that $M(S)$ is arbitrarily small-where M is two-dimensional Lebesgue measure. This solved the most general version of Kakeya's problem, which asked how small $M(S)$ could be if no further restrictions were placed on $S$. But was it ever proved that $M(S)$ could never actually be zero? I ask because Besicovich also proved that there existed subsets T of the Euclidean plane such that $M(T)=0$ and such that pairwise disjoint straight line segments of fixed length, having every possible orientation, were contained in $S$ (as subsets).

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Just a sketch: consider a motion of the needle (unit line segment), (assume) that it can be represented by a continuous and piece-wise $C^1$ curve in the space of Euclidean motions (translations + rotations). Restrict yourself to a $C^1$ portion such that the starting and ending angles are not the same (else there can be no total rotation). By the mean value theorem there exists a point on the path where the rate of change of the angle is non-zero. It should be straight-forward to check that the image of the needle in a neighborhood of that point contains a disk. – Willie Wong Jul 13 '10 at 15:33

A Kakeya needle set cannot be of measure zero (a line segment cannot be rotated continuously within a set of measure 0). See the blog post by Terry Tao.

However, there are sets of measure zero within which a line segment can be rotated by a Baire-1 map (link).

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Thanks very much for providing such a complete and informative response – Garabed Gulbenkian Jul 13 '10 at 19:46
Hey, you're welcome. It wasn't a huge effort, really. – Andrey Rekalo Jul 13 '10 at 20:09
In his "Birds and frogs" Dyson attributes the following to Besicovitch: "Gentlemen. Fifty million English speak English you speak. Hundred and fifty million Russians speak English I speak." – Wadim Zudilin Jul 13 '10 at 22:44
Hi, Wadim. I speak Ukrainian English. – Andrey Rekalo Jul 14 '10 at 10:03

Don't you need a set of positive measure to swing a cat, even through 0.00001 radians? Intuitively you need to include a small sector of a circle to rotate a line segment about any of its points as fixed centre; and while the problem would allow you to rotate about a centre that changes, I'm not seeing how you can do without something resembling a small sector. But perhaps that is the point of what you're asking?

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